1. © 2010 Pearson Education, Inc. All rights reserved
Complex Numbers
SECTION 1.3
Define complex numbers.
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers. 1 2 3 4
© 2010 Pearson Education, Inc. All rights reserved

2. © 2010 Pearson Education, Inc. All rights reserved
Definition of i
The square root of −1 is called i.
The number i is called the imaginary unit.
© 2010 Pearson Education, Inc. All rights reserved

3. © 2010 Pearson Education, Inc. All rights reserved
Complex Numbers
A complex number is a number of the form
where a and b are real numbers and i2 = –1.
The number a is called the real part of z, and we write Re(z) = a.
The number b is called the imaginary part of z and we write Im(z) = b.
© 2010 Pearson Education, Inc. All rights reserved

4. © 2010 Pearson Education, Inc. All rights reserved
Definitions
A complex number z written in the form
a + bi is said to be in standard form.
A complex number with a = 0 and b ≠ 0, written as bi, is called a pure imaginary number.
If b = 0, then the complex number a + bi is a real number. Real numbers form a subset of complex numbers (with imaginary part 0).
© 2010 Pearson Education, Inc. All rights reserved

5. Square Root of a Negative Number
For any positive number, b
© 2010 Pearson Education, Inc. All rights reserved

6. © 2010 Pearson Education, Inc. All rights reserved
Identifying the Real and the Imaginary Parts of a Complex Number
EXAMPLE 1
Identify the real and the imaginary parts of each complex number.
To be discussed in class
© 2010 Pearson Education, Inc. All rights reserved

7. Equality of Complex Numbers
Two complex numbers z = a + bi and w = c + di are equal if and only if
a = c and b = d
That is, z = w if and only if Re(z) = Re(w) and Im(z) = Im(w).
© 2010 Pearson Education, Inc. All rights reserved

8. © 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 2
Equality of Complex Numbers
Find a and b assuming that
(1 – 2a) + 3i = 5 – (2b – 5)i.
Solution
To be solved in class
© 2010 Pearson Education, Inc. All rights reserved

9. ADDITION AND SUBTRACTION OF COMPLEX NUMBERS
For all real numbers a, b, c, and d, let
z = a + bi and w = c + di.
© 2010 Pearson Education, Inc. All rights reserved

10. © 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 3
Adding and Subtracting Complex Numbers
Textbook Exercises Page 115
To be discussed in class
© 2010 Pearson Education, Inc. All rights reserved

11. MULTIPLYING COMPLEX NUMBERS
For all real numbers a, b, c, and d,
© 2010 Pearson Education, Inc. All rights reserved

12. © 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 4
Multiplying Complex Numbers
Textbook Exercises Page 115
To be discussed in class
© 2010 Pearson Education, Inc. All rights reserved

13. CONJUGATE OF A COMPLEX NUMBER
If z = a + bi, then the conjugate (or complex conjugate) of z is denoted by and defined by
© 2010 Pearson Education, Inc. All rights reserved

14. © 2010 Pearson Education, Inc. All rights reserved
Multiplying a Complex Number by Its Conjugate
EXAMPLE 5
Find the product for each complex number .
a. z = 4 + 5i b. z = 1 – 2i
To be discussed in class
© 2010 Pearson Education, Inc. All rights reserved

15. COMPLEX CONJUGATE PRODUCT THEOREM
If z = a + bi, then
© 2010 Pearson Education, Inc. All rights reserved

16. DIVIDING COMPLEX NUMBERS
To write the quotient of two complex numbers w and z (z ≠ 0), and write
and then write the right side in standard form.
© 2010 Pearson Education, Inc. All rights reserved

17. © 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 6
Dividing Complex Numbers
Write the following quotients in standard form.
To be discussed in class
© 2010 Pearson Education, Inc. All rights reserved